Freezer Coils Ice Over and Thermostat Always Has Continuity

I have a GE side by side. My freezer coils ice over about every 1-2 months. It makes the fridge warm. I've been defrosting the coils with a hair dryer, which buys me some time before it occurs again, but I'd like to permanently fix the problem.

I bought a new thermostat (the one that sits at the top of the coils) and wired it in. It'll take some time before the coils either ice over (or not) so I don't know if the thermostat was the problem. Hence, I'd like to test the old thermostat to see if it was defective.

From what I understand, if I check continuity on a thermostat, I should get continuity across the wires if it is cold, but not if it is warm. Well, I've tested my old thermostat while it was freezing and also while it was room temperature. Either way it passes for continuity.

This confuses me, as I assumed that when it passes for continuity, the circuit is complete and the defroster would run. That makes it seem like the defroster would run 24x7, which would mean no frost on the coils (and maybe some other problems). I expected to see no continuity, and therefore no defrosting. Am I understanding this wrong?

I'm not quite sure how to determine if my old thermostat is defective.

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Freezer Coils Ice Over and Thermostat Always Has Continuity

First, I am no expert but someone else grappling with similar problems. But I would guess that one possibility is that giving continuous continuity the defrost heater burned out. If you have a vom you can test it but not sure of details.
You can look for my post if you want to admire someone even deeper in the mire than yourself.